A197870 Expansion of false theta series variation of Ramanujan theta function psi(x).

1, -1, 0, 1, 0, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Michael Somos, Oct 18 2011

Keywords

• sign

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 - x + x^3 - x^6 + x^10 - x^15 + x^21 - x^28 + x^36 - x^45 + x^55 + ...
G.f. = q - q^9 + q^25 - q^49 + q^81 - q^121 + q^169 - q^225 + q^289 - q^361 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A010054.

Cf. A005169 (g.f.: G(0), where G(k) = 1/( 1 - q^(k+1)*G(k+1) ) ).

Programs

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-1)^k x^(k (k + 1)/2), {k, 0, (Sqrt[8 n + 1] - 1)/2}], {x, 0, n}]]; (* Michael Somos, Jul 21 2014 *)
a[n_] := Module[{r, k}, r = Reduce[k >= 0 && 2n == k(k+1), k, Integers]; If[r === False, 0, (-1)^r[[2]] ] ]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Nov 27 2016 *)

PARI

{a(n) = local(x); if( issquare( 8*n + 1, &x), (-1)^(x\2), 0)};

Formula

G.f.: Sum_{k>=0} (-1)^k * x^(k*(k+1)/2). |a(n)| = A010054(n).

G.f.: G(0) where G(k) = 1 - q^(k+1)*G(k+1) = 1 - q*(1 - q^2*(1 - q^3*(1 - q^4*(1 - ...)))). - Joerg Arndt, Jun 29 2013

G.f.: Sum_{k>=0} a(k) * x^(3*k) = 1 / (1+x) + x*(1-x) / ((1+x)*(1+x^2)*(1+x^3)) + x^2*(1-x)*(1-x^3) / ((1+x)*(1+x^2)*...*(1+x^5)) + ... - Michael Somos, Jul 21 2014